High-Rank Matrix Completion and Clustering under Self-Expressive Models

We propose efficient algorithms for simultaneous clustering and completion of incomplete high-dimensional data that lie in a union of low-dimensional subspaces. We cast the problem as finding a completion of the data matrix so that each point can be reconstructed as a linear or affine combination of a few data points. Since the problem is NP-hard, we propose a lifting framework and reformulate the problem as a group-sparse recovery of each incomplete data point in a dictionary built using incomplete data, subject to rank-one constraints. To solve the problem efficiently, we propose a rank pursuit algorithm and a convex relaxation. The solution of our algorithms recover missing entries and provides a similarity matrix for clustering. Our algorithms can deal with both low-rank and high-rank matrices, does not suffer from initialization, does not need to know dimensions of subspaces and can work with a small number of data points. By extensive experiments on synthetic data and real problems of video motion segmentation and completion of motion capture data, we show that when the data matrix is low-rank, our algorithm performs on par with or better than low-rank matrix completion methods, while for high-rank data matrices, our method significantly outperforms existing algorithms.